DO NOT WASTE MY QUESTION, ANSWER $*$ALL$*$ SECThe vibration in the vertical direction of an airplane and its wings can be modeled as

a three$-$degree$-$of$-$freedom system with one mass corresponding to the right wing, one

mass for the left wing, and one mass for the fuselage. The stiffness connecting the

three masses corresponds to that of the wing and is a function of the modulus $E$ of

the wing, moment of inertia I, and length $L.$ Consider a gust of wind $u(t)$ applied

upwards onto the right wing. We are interested in the time response of the aircraft

when experiencing such a wind disturbance.

Figure $2$: Simplified model of an aircraft and its wings.

$($a$)$ What are the equations of motion for the fuselage and each wing? $($Ignore gravi$-$

tational effects.$)$

$($b$)$ The equations of motion can be expressed in the following matrix$-$vector form:

$m\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_l^{}\\ x_f^{}\\ x_r^{}\end{array}\right]+\frac{3EI}{L^3}\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]\left[\begin{array}{c}{x}_l\\ x_f\\ x_r\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u.$

$($Feel free to verify this.$)$ Suppose you can only observe $x_f.$ What is the equivalent

state space dynamics of the system $x^{}=Ax+Bu,y=Cx(D=0).$ What is the

state vector $x,$ system matrix $A,$ input matrix $B,$ and observability matrix $C?$

$($c$)$ With the programming language of your choice, simulate the impulse response

for the fuselage for $20$ seconds. Let $E=6\mathrm{.}9{10}^{9}N{m}^{-2},L=2m,m=3000kg,$

$I=5\mathrm{.}2{10}^{-6}{m}^4.$ Include your plots and briefly comment on the response.

Include a copy of your code.TIONS A$-$C FOR AN UPVOTE. SHOW ALL WORK AND STEPS, SHOW FULL CALCULATIONS. WILL RATE IMMEDIATELY!!